I need help proving this conjecture:
Suppose you have two Natural Numbers, n and m, such that n = m + 2. Then 1/n + 1/m is equal to some fraction which, when reduced, is equal to $\frac ab$ (for some a,b in the natural numbers) such that $a^2 + b^2 = c^2$ for some c in the natural numbers.
We have $\frac{1}{n}+\frac{1}{n+2}=\frac{2n+2}{n(n+2)}$.
Note that actually you do not need to have a reduced fraction. If the numerator and the denominator have a common factor just cancel it when you want.
We can see that $a^2+b^2=(2n+2)^2+(n(n+2))^2=n^4+4n^3+8n^2+8n+4=(n^2+2n+2)^2=c^2$
This proves your conjecture.